80
McDonald •
Fundamentals of Derivatives Markets
The one year implied forward rate from Year 3 to Year 4 is determined by Equation (7.3):
0
(0, 3)
0.8763
(3, 4)
1
1
0.06515
(0, 4)
0.8227
P
r
P
 =
 =
The par coupon rate for a four year maturity uses Equation (7.6) with
0,
t
=
4,
T
=
and four yearly coupons:
1
(0, 4)
(0,1)
(0, 2)
(0, 3)
(0, 4)
1 0.8227
0.96154
0.91573
0.8763
0.8227
0.04958
P
c
P
P
P
P

=
+
+
+

=
+
+
+
=
The four year continuous yield is:
4
ln(0.8227)
0.04879
4
r

=
=
n
Question 7.2
The coupon bond pays a coupon of $60 each year plus the principal of $1,000 after five years. We have
cash flows of
60, 60, 60, 60,1060.
To obtain the price of the coupon bond, we multiply each cash flow by
the zerocoupon bond price of that year. Specifically,
Bond Price
60 ( (0,1)
(0, 2)
(0, 3)
(0, 4)
(0, 5)) 1000
(0, 5)
60 (0.96154
0.91573
0.87630
0.8227
0.77611) 1000 0.77611
60 4.3524
776.11
261.14
776.11 1037.25
P
P
P
P
P
P
=
×
+
+
+
+
+
×
=
×
+
+
+
+
+
×
=
×
+
=
+
=
This yields a bond price of $1,037.25.
n
Question 7.3
This is a straightforward application of Equations (7.1), (7.3), and (7.6). We also need the continuous rate
calculation (derived in Question 7.1):
ln(1/ (0, ))
ln( (0, ))
(0, )
cc
P
T
P
T
r
T
T
T

=
=
Maturity
ZeroCoupon
Bond Yield
Zero Coupon
Bond Price
OneYear Implied
Forward Rate
Par
Coupon
Cont. Comp.
Zero Yield
1
0.03000
0.97087
0.03000
0.03000
0.02956
2
0.03500
0.93351
0.04002
0.03491
0.03440
3
0.04000
0.88900
0.05007
0.03974
0.03922
4
0.04500
0.83856
0.06014
0.04445
0.04402
5
0.05000
0.78353
0.07024
0.04903
0.04879